User Guide for the TIMSS International Database.pdf - TIMSS and ...

C H A P T E R 5 S C A L I N G
Likewise, the scoring function of response k to item i is a scalar, b ik, in the former, whereas it is
a D by 1 column vector, b ik, in the latter.
5.4 The Population Model
The item response model is a conditional model in the sense that it describes the process of
generating item responses conditional on the latent variable, q. The complete definition of the
TIMSS model, therefore, requires the specification of a density, fq( qa ; ), for the latent
variable, q. We use a to symbolize a set of parameters that characterize the distribution of q.
The most common practice when specifying unidimensional marginal item response models
is to assume that the students have been sampled from a normal population with mean m and
variance s 2 . That is:
( )
fqf qa ; ; , exp
( ) º 2
q ( qms)
=
1
2
2ps
2
é q - m ù
ê-
2 ú
ëê
2s
ûú
(8)
or equivalently q = m+E
(9)
where
2
E ~ N(
0,
s ).
Adams, Wilson, and Wu (1997) discuss how a natural extension of (8) is to replace the mean,
T m, with the regression model, Yn b , where Y is a vector of u fixed and known values for
n
student n, and b is the corresponding vector of regression coefficients. For example, Yn could
be constituted of student variables such as gender, socio-economic status, or major. Then the
population model for student n, becomes
T
qn = Ynb+ En
, (10)
where we assume that the E are independently and identically normally distributed with mean
n
zero and variance s 2 so that (10) is equivalent to
T T
T
fq qn; Yn, , s ps exp qn — Yn qn
— Yn
s b
2 2 -12
1
( ) = 2
é
( ) - 2 ( b) ( b
ù ) , (11)
ëê 2
ûú
T 2
a normal distribution with mean Yn b and variance s . If (11) is used as the population model
then the parameters to be estimated are b, s 2 , and x.
The TIMSS scaling model takes the generalization one step further by applying it to the
vector valued q rather than the scalar valued q resulting in the multivariate population model
-d
2 -1
T -1
fq( qn; Wn, g, S) = S exp
é 1
( 2p
2 ) - ( qn — gWn) S ( qn — gW
ù
n)
, (12)
ëê 2
ûú
where g is a u ´ d matrix of regression coefficients, S is a d ´ d variance-covariance matrix
and Wn is a u ´ 1 vector of fixed variables. If (12) is used as the population model then the
parameters to be estimated are g, S, and x.
In TIMSS we refer to the W n variables as conditioning variables.
5 - 4 T I M S S D A T A B A S E U S E R G U I D E